Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvrelsymb Structured version   Visualization version   GIF version

Theorem eqvrelsymb 35881
 Description: An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised and distinct variable conditions removed by Peter Mazsa, 2-Jun-2019.)
Hypothesis
Ref Expression
eqvrelsymb.1 (𝜑 → EqvRel 𝑅)
Assertion
Ref Expression
eqvrelsymb (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem eqvrelsymb
StepHypRef Expression
1 eqvrelsymb.1 . . . 4 (𝜑 → EqvRel 𝑅)
21adantr 484 . . 3 ((𝜑𝐴𝑅𝐵) → EqvRel 𝑅)
3 simpr 488 . . 3 ((𝜑𝐴𝑅𝐵) → 𝐴𝑅𝐵)
42, 3eqvrelsym 35880 . 2 ((𝜑𝐴𝑅𝐵) → 𝐵𝑅𝐴)
51adantr 484 . . 3 ((𝜑𝐵𝑅𝐴) → EqvRel 𝑅)
6 simpr 488 . . 3 ((𝜑𝐵𝑅𝐴) → 𝐵𝑅𝐴)
75, 6eqvrelsym 35880 . 2 ((𝜑𝐵𝑅𝐴) → 𝐴𝑅𝐵)
84, 7impbida 800 1 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   class class class wbr 5039   EqvRel weqvrel 35510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-refrel 35792  df-symrel 35820  df-trrel 35850  df-eqvrel 35860 This theorem is referenced by:  eqvrelth  35886
 Copyright terms: Public domain W3C validator